Abelian Divisibility Sequences Joseph H. Silverman Brown University - - PowerPoint PPT Presentation

Abelian Divisibility Sequences

Joseph H. Silverman

Brown University Arithmetic of Low-Dimensional Abelian Varietes

ICERM, June 3–7 2019

Classical Divisibility Sequences

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Divisibility Sequences A divisibility sequence is a sequence of (positive) integers (Dn)n≥1 such that m | n = ⇒ Dm | Dn.

Classical Divisibility Sequences

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Divisibility Sequences A divisibility sequence is a sequence of (positive) integers (Dn)n≥1 such that m | n = ⇒ Dm | Dn. Classical examples divisibility sequences include: Dn = an − bn, where a > b ≥ 1; the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, . . . .

Classical Divisibility Sequences

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Divisibility Sequences A divisibility sequence is a sequence of (positive) integers (Dn)n≥1 such that m | n = ⇒ Dm | Dn. Classical examples divisibility sequences include: Dn = an − bn, where a > b ≥ 1; the Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, . . . . An elliptic divisibility sequence (EDS) is formed from an elliptic curve E/Q and a non-torsion point P ∈ E(Q) by writing nP = An(P) Dn(P)2, Bn(P) Dn(P)3

• .

The sequence

• Dn(P)
• n≥1 is an EDS.

Classical Divisibility Sequences

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Divisibility Sequences over Dedekind Domains More generally, if R is a Dedekind domain, we define an R-divisibility sequence to be a sequence of ideals (Dn)n≥1 such that m | n = ⇒ Dm | Dn.

Classical Divisibility Sequences

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Divisibility Sequences over Dedekind Domains More generally, if R is a Dedekind domain, we define an R-divisibility sequence to be a sequence of ideals (Dn)n≥1 such that m | n = ⇒ Dm | Dn. In this way we can define an EDS, for example, by fac- toring the ideal generated by x(nP) in the form x(nP)R = An(P)Dn(P)−2 and taking the sequence

• Dn(P)
• n≥1.

Classical Divisibility Sequences

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Reformulating EDS Let E/K be an elliptic curve, let P ∈ E(K), and let E/R be a N´ eron model for E/K. Then the EDS

• Dn(P)
• n≥1

is characterized by noting that for each prime ideal p

• f R, we have∗
• rdp Dn(P) =
• largest k so that nP ≡ O (mod pk)
• .

∗ Maybe not quite right at primes of bad reduction.

Classical Divisibility Sequences

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Reformulating EDS Let E/K be an elliptic curve, let P ∈ E(K), and let E/R be a N´ eron model for E/K. Then the EDS

• Dn(P)
• n≥1

is characterized by noting that for each prime ideal p

• f R, we have∗
• rdp Dn(P) =
• largest k so that nP ≡ O (mod pk)
• .

∗ Maybe not quite right at primes of bad reduction.

Or we can simply say that Dn(P) is the largest ideal (ordered by divisibility) such that nP ≡ O (mod Dn(P)).

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type I In general: R a Dedekind domain. K the fraction field of R. A/K an abelian variety. A/R a N´ eron model for A/K. P ∈ A(K) a non-torsion point. The abelian divisibility sequence (ADS) for the pair (A, P) is the sequence of ideals

• Dn(P)
• n≥1 defined

by the property that Dn(P) is the largest ideal satisfying nP ≡ O (mod Dn(P)).

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type I In general: R a Dedekind domain. K the fraction field of R. A/K an abelian variety. A/R a N´ eron model for A/K. P ∈ A(K) a non-torsion point. The abelian divisibility sequence (ADS) for the pair (A, P) is the sequence of ideals

• Dn(P)
• n≥1 defined

by the property that Dn(P) is the largest ideal satisfying nP ≡ O (mod Dn(P)). Alternatively, letting π : A → Spec(R), we can de- fine Dn(P) via arithmetic intersection theory, Dn(P) := π∗(nP · O) = (nP)∗(O) ∈ Div

• Spec(R)
• .

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type I In general: R a Dedekind domain. K the fraction field of R. A/K an abelian variety. A/R a N´ eron model for A/K. P ∈ A(K) a non-torsion point. The abelian divisibility sequence (ADS) for the pair (A, P) is the sequence of ideals

• Dn(P)
• n≥1 defined

by the property that Dn(P) is the largest ideal satisfying nP ≡ O (mod Dn(P)). Alternatively, letting π : A → Spec(R), we can de- fine Dn(P) via arithmetic intersection theory, Dn(P) := π∗(nP · O) = (nP)∗(O) ∈ Div

• Spec(R)
• .

Exercise: Prove that

• Dn(P)
• is a divisibility sequence.

Classical Divisibility Sequences

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Growth Rates A Gm-divisibility sequence D = (Dn) such as an − bn

• r the Fibonacci sequence grows exponentially,

lim

n→∞ |Dn|1/n > 1.

Classical Divisibility Sequences

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Growth Rates A Gm-divisibility sequence D = (Dn) such as an − bn

• r the Fibonacci sequence grows exponentially,

lim

n→∞ |Dn|1/n > 1.

Elliptic divisiblity sequences D =

• Dn(P)
• grow even

faster, lim

n→∞

• NK/Q Dn(P)

1/n2 > 1. (∗) Two remarks about elliptic divisibility sequences:

• The limit in (∗) is ˆ

HE(P), i.e., NK/Q Dn(P) ≈ ˆ HE(P)n2 = ˆ HE(nP).

• The proof uses a deep, ineffective theorem of Siegel.

Classical Divisibility Sequences

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Growth Rates A Gm-divisibility sequence D = (Dn) such as an − bn

• r the Fibonacci sequence grows exponentially,

lim

n→∞ |Dn|1/n > 1.

Elliptic divisiblity sequences D =

• Dn(P)
• grow even

faster, lim

n→∞

• NK/Q Dn(P)

1/n2 > 1. (∗) Two remarks about elliptic divisibility sequences:

• The limit in (∗) is ˆ

HE(P), i.e., NK/Q Dn(P) ≈ ˆ HE(P)n2 = ˆ HE(nP).

• The proof uses a deep, ineffective theorem of Siegel.

The height of nP on an abelian variety grows at a sim- ilar rate, but co-dimension considerations suggest that Dn(P) might not grow that fast.

Classical Divisibility Sequences

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Growth Rates of ADS: A Conjecture Conjecture 1. Let A/K be an abelian variety of dimension ≥ 2, and let P ∈ A(K) be a point such that ZP is Zariski dense in A. Then lim

n→∞

• NK/Q Dn(P)

1/n2 = 1.

Classical Divisibility Sequences

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Growth Rates of ADS: A Conjecture Conjecture 1. Let A/K be an abelian variety of dimension ≥ 2, and let P ∈ A(K) be a point such that ZP is Zariski dense in A. Then lim

n→∞

• NK/Q Dn(P)

1/n2 = 1. The conjecture says that in dimension ≥ 2, an ADS grows more slowly than the heights of the points in the sequence nP.

Classical Divisibility Sequences

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Growth Rates of ADS: A Conjecture Conjecture 1. Let A/K be an abelian variety of dimension ≥ 2, and let P ∈ A(K) be a point such that ZP is Zariski dense in A. Then lim

n→∞

• NK/Q Dn(P)

1/n2 = 1. The conjecture says that in dimension ≥ 2, an ADS grows more slowly than the heights of the points in the sequence nP.

• Theorem. Conjecture 1 follows from Vojta’s conjec-

ture applied to A blown up at O.

Classical Divisibility Sequences

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A Multiplicative Analogue to Conjecture 1 Here is a Gm analogue. We replace A by G2

m and P ∈

A(K) with (a, b) ∈ G2

m(Q). The associated divisibility

sequence gcd(an − 1, bn − 1) measures the “arithmetic distance” from (a, b)n to (1, 1).

Classical Divisibility Sequences

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A Multiplicative Analogue to Conjecture 1 Here is a Gm analogue. We replace A by G2

m and P ∈

A(K) with (a, b) ∈ G2

m(Q). The associated divisibility

sequence gcd(an − 1, bn − 1) measures the “arithmetic distance” from (a, b)n to (1, 1). Theorem. (Bugeaud–Corvaja–Zannier 2003) Let a, b ∈ Z with |a| > |b| > 1. Then lim

n→∞ gcd(an − 1, bn − 1)1/n = 1.

([BCZ] result is more general. See also work of A. Levin.)

Classical Divisibility Sequences

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A Multiplicative Analogue to Conjecture 1 Here is a Gm analogue. We replace A by G2

m and P ∈

A(K) with (a, b) ∈ G2

m(Q). The associated divisibility

sequence gcd(an − 1, bn − 1) measures the “arithmetic distance” from (a, b)n to (1, 1). Theorem. (Bugeaud–Corvaja–Zannier 2003) Let a, b ∈ Z with |a| > |b| > 1. Then lim

n→∞ gcd(an − 1, bn − 1)1/n = 1.

([BCZ] result is more general. See also work of A. Levin.) The proof uses Schmidt’s subspace theorem and is sur- prisingly intricate, even for a = 3 and b = 2. Challenge: Give an elementary proof that gcd(3n − 1, 2n − 1)1/n − → 1.

Classical Divisibility Sequences

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Growth Rates of ADS: Another Conjecture Conjecture 1 says that an ADS does not grow too fast. The next conjecture says that for many n, it doesn’t grow at all!

Classical Divisibility Sequences

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Growth Rates of ADS: Another Conjecture Conjecture 1 says that an ADS does not grow too fast. The next conjecture says that for many n, it doesn’t grow at all! Conjecture 2. Let A/K be an abelian variety of dimension ≥ 2, and let P ∈ A(K) be a point such that ZP is Zariski dense in A. Then there is a constant C = C(A/K, P) with the property that (∗)

• NK/Q Dn(P)
• ≤ C for infinitely many n ≥ 1.

Classical Divisibility Sequences

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Growth Rates of ADS: Another Conjecture Conjecture 1 says that an ADS does not grow too fast. The next conjecture says that for many n, it doesn’t grow at all! Conjecture 2. Let A/K be an abelian variety of dimension ≥ 2, and let P ∈ A(K) be a point such that ZP is Zariski dense in A. Then there is a constant C = C(A/K, P) with the property that (∗)

• NK/Q Dn(P)
• ≤ C for infinitely many n ≥ 1.

Bolder Conjecture: The set of primes n such that the inequality (∗) holds is a set of positive density.

Classical Divisibility Sequences

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Growth Rates of ADS: Another Conjecture Conjecture 1 says that an ADS does not grow too fast. The next conjecture says that for many n, it doesn’t grow at all! Conjecture 2. Let A/K be an abelian variety of dimension ≥ 2, and let P ∈ A(K) be a point such that ZP is Zariski dense in A. Then there is a constant C = C(A/K, P) with the property that (∗)

• NK/Q Dn(P)
• ≤ C for infinitely many n ≥ 1.

Bolder Conjecture: The set of primes n such that the inequality (∗) holds is a set of positive density. Even Bolder Conjecture Question: The set of primes n such that the inequality (∗) holds is a set of density 1.

Classical Divisibility Sequences

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A Multiplicative Analogue to Conjecture 2 It is surprising to me that this conjecture wasn’t formu- lated until quite recently.

• Conjecture. (Ailon–Rudnick 2004) Let a, b ∈ Z with

|a| > |b| > 1. Then there are infinitely many values

• f n ≥ 1 such that

gcd(an − 1, bn − 1) = gcd(a − 1, b − 1).

Classical Divisibility Sequences

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A Multiplicative Analogue to Conjecture 2 It is surprising to me that this conjecture wasn’t formu- lated until quite recently.

• Conjecture. (Ailon–Rudnick 2004) Let a, b ∈ Z with

|a| > |b| > 1. Then there are infinitely many values

• f n ≥ 1 such that

gcd(an − 1, bn − 1) = gcd(a − 1, b − 1). Even for gcd(3n − 1, 2n − 1), there seem to be no tools to attack the problem. However, we do have:

Classical Divisibility Sequences

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A Multiplicative Analogue to Conjecture 2 It is surprising to me that this conjecture wasn’t formu- lated until quite recently.

• Conjecture. (Ailon–Rudnick 2004) Let a, b ∈ Z with

|a| > |b| > 1. Then there are infinitely many values

• f n ≥ 1 such that

gcd(an − 1, bn − 1) = gcd(a − 1, b − 1). Even for gcd(3n − 1, 2n − 1), there seem to be no tools to attack the problem. However, we do have:

• Theorem. (Ailon–Rudnick 2004) Let a(T), b(T) ∈

C[T] be multiplicatively independent modulo C∗. Then there is a c(T) ∈ C[t] such that gcd

• a(T)n − 1, b(T)n − 1

c(T) for all n ≥ 1.

Classical Divisibility Sequences

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Experiments I’ve gathered a fair amount of data for the G2

m conjec-

ture, i.e., Ailon–Rudnick’s conjecture for gcd(an − 1, bn − 1) = gcd(a − 1, b − 1), which I will display on the next slide.

Classical Divisibility Sequences

10

Experiments I’ve gathered a fair amount of data for the G2

m conjec-

ture, i.e., Ailon–Rudnick’s conjecture for gcd(an − 1, bn − 1) = gcd(a − 1, b − 1), which I will display on the next slide. I’ve also gathered some data for Conjecture 2 when A = E1 × E2 is a product of elliptic curves.

Classical Divisibility Sequences

10

Experiments I’ve gathered a fair amount of data for the G2

m conjec-

ture, i.e., Ailon–Rudnick’s conjecture for gcd(an − 1, bn − 1) = gcd(a − 1, b − 1), which I will display on the next slide. I’ve also gathered some data for Conjecture 2 when A = E1 × E2 is a product of elliptic curves. It would be interesting to do some experiments using simple abelian surfaces.

0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1000 2000 3000 4000 5000 6000 Frequency Number of primes p Frequency of gcd(ap - 1, bp - 1) = gcd(a - 1,b - 1), p prime

a=2, b=3 a=2, b=5 a=2, b=7 a=3, b=5 a=3, b=7

Question: Does the frequency go to 100%?

Classical Divisibility Sequences

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Fast Growth Versus Slow Growth As we have seen the growth rate of the ADS associated to an abelian variety A is Fast if dim(A) = 1; Slow if dim(A) ≥ 2 (conjecturally).

Classical Divisibility Sequences

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Fast Growth Versus Slow Growth As we have seen the growth rate of the ADS associated to an abelian variety A is Fast if dim(A) = 1; Slow if dim(A) ≥ 2 (conjecturally). There are a number of reasons why fast-growing divisi- bility sequences are useful, including:

• Existence of primitive prime divisors (defined later);
• Applications to logic, Hilbert’s 10th problem;
• Applications to cryptography based on discrete loga-

rithms and/or pairings.

Classical Divisibility Sequences

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Fast Growth Versus Slow Growth As we have seen the growth rate of the ADS associated to an abelian variety A is Fast if dim(A) = 1; Slow if dim(A) ≥ 2 (conjecturally). There are a number of reasons why fast-growing divisi- bility sequences are useful, including:

• Existence of primitive prime divisors (defined later);
• Applications to logic, Hilbert’s 10th problem;
• Applications to cryptography based on discrete loga-

rithms and/or pairings. How can we get fast-growing, geometrically defined, abelian divisibility sequences in high dimension?

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type II Let π : A → Spec(R) be a N´ eron model for A/K. For a point P ∈ A(K) on the generic fiber, let P ⊂ A be the closure of P. We defined the Type I ADS for (A, P) to be the sequence of ideals Dn(P) := π∗

• [n]P · O
• = π∗
• P · [n]∗O
• .

This is small when dim(A) ≥ 2 because dim(A) ≥ 3, while dim P = dim O = 1.

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type II Let π : A → Spec(R) be a N´ eron model for A/K. For a point P ∈ A(K) on the generic fiber, let P ⊂ A be the closure of P. We defined the Type I ADS for (A, P) to be the sequence of ideals Dn(P) := π∗

• [n]P · O
• = π∗
• P · [n]∗O
• .

This is small when dim(A) ≥ 2 because dim(A) ≥ 3, while dim P = dim O = 1. To obtain a fast-growing sequence, we should replace P and/or O with a higher-dimensional variety.

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type II Let π : A → Spec(R) be a N´ eron model for A/K. For a point P ∈ A(K) on the generic fiber, let P ⊂ A be the closure of P. We defined the Type I ADS for (A, P) to be the sequence of ideals Dn(P) := π∗

• [n]P · O
• = π∗
• P · [n]∗O
• .

This is small when dim(A) ≥ 2 because dim(A) ≥ 3, while dim P = dim O = 1. To obtain a fast-growing sequence, we should replace P and/or O with a higher-dimensional variety. And in order to get a divisibility sequence, we need to replace P, not O.

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type II Definition: Let X ⊂ AK be an irreducible codimen- sion 1 subvariety defined over K, and let X be its clo- sure in A. The abelian divisibility sequence for the pair (A, X) is the sequence of ideals∗ Dn(X) := π∗

• X · [n]∗O
• .

∗ Need to be a bit careful if X contains a component of [n]∗O.

Classical Divisibility Sequences

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Abelian Divisibility Sequences: Type II Definition: Let X ⊂ AK be an irreducible codimen- sion 1 subvariety defined over K, and let X be its clo- sure in A. The abelian divisibility sequence for the pair (A, X) is the sequence of ideals∗ Dn(X) := π∗

• X · [n]∗O
• .

∗ Need to be a bit careful if X contains a component of [n]∗O.

Conjecture. If A is simple, or more generally if X contains no translates of abelian subvarieties, then Dn(X) is fast-growing: lim inf

n→∞

• NK/Q Dn(X)
• 1/n2 dim A

> 1.

Classical Divisibility Sequences

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Tori Divisibility Sequences: Type II The analogous problem for GN

m is solved.

Theorem. (Habegger, Dimitrov, 2016) Let f ∈ R[T ±1

1

, . . . , T ±1

N ] be a Laurent polynomial, and let

Xf ⊂ GN

m be the associated divisor. Then

lim

n→∞

• NK/Q Dn(Xf)
• 1/nN

= MahlerMeasure(f).

Classical Divisibility Sequences

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Tori Divisibility Sequences: Type II The analogous problem for GN

m is solved.

Theorem. (Habegger, Dimitrov, 2016) Let f ∈ R[T ±1

1

, . . . , T ±1

N ] be a Laurent polynomial, and let

Xf ⊂ GN

m be the associated divisor. Then

lim

n→∞

• NK/Q Dn(Xf)
• 1/nN

= MahlerMeasure(f). Dn(Xf) :=

• ζ1,...,ζN∈µn

f(ζ1, . . . , ζN).

Classical Divisibility Sequences

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Tori Divisibility Sequences: Type II The analogous problem for GN

m is solved.

Theorem. (Habegger, Dimitrov, 2016) Let f ∈ R[T ±1

1

, . . . , T ±1

N ] be a Laurent polynomial, and let

Xf ⊂ GN

m be the associated divisor. Then

lim

n→∞

• NK/Q Dn(Xf)
• 1/nN

= MahlerMeasure(f). Dn(Xf) :=

• ζ1,...,ζN∈µn

f(ζ1, . . . , ζN).

• Remark. The theorem is false if we allow f to have C
• coefficients. Even for N = 1, the theorem requires some

sort of estimate coming from linear-forms-in-logarithms, because we need to know that algebraic numbers cannot be too closely approximated by roots of unity.

Classical Divisibility Sequences

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Primitive Prime Divisors and Zsigmondy Sets Question: Which terms contain a “new” prime divisor? Definition: Let D := (Dn)n≥1 be a sequence of ideals. A primitive prime divisor of Dn is a prime ideal p satisfying p | Dn and p ∤ Dm for all m < n.

Classical Divisibility Sequences

16

Primitive Prime Divisors and Zsigmondy Sets Question: Which terms contain a “new” prime divisor? Definition: Let D := (Dn)n≥1 be a sequence of ideals. A primitive prime divisor of Dn is a prime ideal p satisfying p | Dn and p ∤ Dm for all m < n. The Zsigmondy set of D specifies the terms that do not have a primitive prime divisor: Z(D) := {n ≥ 1 : Dn has no primitive prime divisors}.

Classical Divisibility Sequences

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Primitive Prime Divisors and Zsigmondy Sets Question: Which terms contain a “new” prime divisor? Definition: Let D := (Dn)n≥1 be a sequence of ideals. A primitive prime divisor of Dn is a prime ideal p satisfying p | Dn and p ∤ Dm for all m < n. The Zsigmondy set of D specifies the terms that do not have a primitive prime divisor: Z(D) := {n ≥ 1 : Dn has no primitive prime divisors}. Some Sample Results:

• Bang/Zsigmondy (1886/92): Z(an − bn) ⊆ {1, 2, 6}.
• Carmichael (1913): Z(Fibonacci sequence) = {1, 2, 6, 12}.
• Bilu–Hanrot–Voutier (2001):

Z(Lucas/Lehmer sequence) ⊂ {1, 2, . . . , 30}.

• JS (1988): Z(elliptic divisibility sequence) is finite.

Classical Divisibility Sequences

17

Primitive Prime Divisors in Abelian Divisibility Sequences There are two ingredients that go into proofs that the Zsigmondy set of a sequence D = (Dn)n≥1 is finite:

• The sequence grows rapidly in norm, e.g.,

log NK/Q Dn ≫ nδ for some δ > 1.

• The p-divisiblity does not grow too rapidly, e.g., let r

be the smallest index with p | Dr, then

• rdp Dnr = ordp Dr + O
• rdp(n)
• .

Classical Divisibility Sequences

17

Primitive Prime Divisors in Abelian Divisibility Sequences There are two ingredients that go into proofs that the Zsigmondy set of a sequence D = (Dn)n≥1 is finite:

• The sequence grows rapidly in norm, e.g.,

log NK/Q Dn ≫ nδ for some δ > 1.

• The p-divisiblity does not grow too rapidly, e.g., let r

be the smallest index with p | Dr, then

• rdp Dnr = ordp Dr + O
• rdp(n)
• .

For ADS of Type I, the Ailon–Rudnick conjecture implies that Z(D) is infinite.

Classical Divisibility Sequences

17

Primitive Prime Divisors in Abelian Divisibility Sequences There are two ingredients that go into proofs that the Zsigmondy set of a sequence D = (Dn)n≥1 is finite:

• The sequence grows rapidly in norm, e.g.,

log NK/Q Dn ≫ nδ for some δ > 1.

• The p-divisiblity does not grow too rapidly, e.g., let r

be the smallest index with p | Dr, then

• rdp Dnr = ordp Dr + O
• rdp(n)
• .

For ADS of Type I, the Ailon–Rudnick conjecture implies that Z(D) is infinite. For ADS of Type II, we conjecturally have rapid norm growth, but p-divisibility when dim ≥ 2 is much less regular than for dim = 1. Again, I’ve done experiments and have weak partial results for G2

m, but it would be

very interesting to gather data for abelian surfaces.

Classical Divisibility Sequences

18

Primes in Divisibility Sequences Let D = (Dn)n≥1 be a divisibility sequence in Z. Natual Question: Are there infinitely many primes in the sequence |Dn/D1|?

Classical Divisibility Sequences

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Primes in Divisibility Sequences Let D = (Dn)n≥1 be a divisibility sequence in Z. Natual Question: Are there infinitely many primes in the sequence |Dn/D1|? Since Dk | Dmk, should consider |Dp/D1| with p prime.

Classical Divisibility Sequences

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Primes in Divisibility Sequences Let D = (Dn)n≥1 be a divisibility sequence in Z. Natual Question: Are there infinitely many primes in the sequence |Dn/D1|? Since Dk | Dmk, should consider |Dp/D1| with p prime. Three Guesses:

• For a Gm-sequence, which typically satisfies

log |Dn| ≫≪ n, we expect Dp/D1 to be prime for infinitely many p. Example: Mersenne sequence Dn = 2n − 1.

• For an EDS , which typically satisfies

log |Dn| ≫≪ n2, we expect Dp/D1 to be prime for only finitely many p.

• Ditto for higher dimensional Type II ADS with

log |Dn| ≫≪ nd.

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19

I want to thank the organizers, Jenn, Noam, Brendan, Bjorn, Drew, and John, for inviting me to speak, and to thank you for your attention.

Abelian Divisibility Sequences

Joseph H. Silverman

Brown University Arithmetic of Low-Dimensional Abelian Varietes

ICERM, June 3–7 2019